By Myron W. Evans, Ilya Prigogine, Stuart A. Rice
The recent variation will give you the sole finished source on hand for non-linear optics, together with exact descriptions of the advances over the past decade from world-renowned specialists.
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Extra info for Advances in Chemical Physics, Vol.119, Part 1. Modern Nonlinear Optics (Wiley 2001)
12 clouds of 1000 points for the same values of the evolution time as in Fig. 11 for Na ¼ 10. Why is the Q function reproduced so well with the classical trajectories? The Q function is a representative of a whole class of the quasidistribution functions. Generally, the s-parametrized quasiprobability distribution for a coherent state, defined by (30), is given by 1 2 2 2 exp À ja À a0 j W ðaÞ ¼ p1 À s 1Às ðsÞ ð142Þ which, for s < 1, is a Gaussian distribution. For s ¼ 1, the distribution becomes the Dirac delta function, for s ¼ 0 it is the Wigner function, and for s ¼ À1, we have the Q function.
The approximate solution correctly predicts the transition from the second harmonic regime to the downconversion regime, which is the physical reason for starting oscillations. The quantum noise really induces macroscopic revivals, but subsequent maxima are smaller and smaller and the second harmonic intensity asymptotically approaches a certain value. Without quantum fluctuations the solution is a monotonic function as shown in the figure by the dotted curve. The quantum noise is necessary to trigger the macroscopic changes in the intensity of the second-harmonic mode.
It was shown by Ou  that the two systems can be solved analytically, giving pﬃﬃﬃ ^ a ð0Þð1 À t tanh tÞ sech t À ÁP ^ a ðtÞ ¼ ÁQ ^ b ð0Þ 2 tanh t sech t ÁQ ^ a ð0Þ p1ﬃﬃﬃ ðtanh t þ t sech2 tÞ þ ÁP ^ b ð0Þ sech2 t ^ b ðtÞ ¼ ÁQ ÁP 2 ^ b ð0Þ p1ﬃﬃﬃ ðsinh t þ t sech tÞ ^ a ð0Þ sech t þ ÁQ ^ a ðtÞ ¼ ÁP ÁP 2 pﬃﬃﬃ ^ b ð0Þð1 À t tanh tÞ ^ b ðtÞ ¼ ÀÁP ^ a ð0Þ 2 tanh t þ ÁQ ÁQ ð86Þ 24 ryszard tanas´ Now, assuming that the two modes are not correlated at time t ¼ 0, it is straightforward to calculate the variances of the quadrature field operators and check, according to the definition (12), whether the field is in a squeezed state.